We introduce a formal ħ\documentclass[12pt]{minimal}
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\begin{document}$$\hbar $$\end{document}-differential operator Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document} that generates higher Koszul brackets on the algebra of (pseudo)differential forms on a P∞\documentclass[12pt]{minimal}
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\begin{document}$$P_{\infty }$$\end{document}-manifold. Such an operator was first mentioned by Khudaverdian and Voronov in arXiv:1808.10049. (This operator is an analogue of the Koszul–Brylinski boundary operator ∂P\documentclass[12pt]{minimal}
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\begin{document}$$\partial _P$$\end{document} which defines Poisson homology for an ordinary Poisson structure.) Here, we introduce Δ=ΔP\documentclass[12pt]{minimal}
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\begin{document}$$\Delta =\Delta _P$$\end{document} by a different method and establish its properties. We show that this BV type operator generating higher Koszul brackets can be included in a one-parameter family of BV type formal ħ\documentclass[12pt]{minimal}
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\begin{document}$$\hbar $$\end{document}-differential operators, which can be understood as a quantization of the cotangent L∞\documentclass[12pt]{minimal}
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\begin{document}$$L_{\infty }$$\end{document}-bialgebroid. We obtain symmetric description on both ΠTM\documentclass[12pt]{minimal}
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\begin{document}$$\Pi TM$$\end{document} and ΠT∗M\documentclass[12pt]{minimal}
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\begin{document}$$\Pi T^*M$$\end{document}. For the purpose of the above, we develop in detail a theory of formal ħ\documentclass[12pt]{minimal}
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\begin{document}$$\hbar $$\end{document}-differential operators and also of operators acting on densities on dual vector bundles. In particular, we have a statement about operators that can be seen as a quantization of the Mackenzie–Xu canonical diffeomorphism. Another interesting feature is that we are able to introduce a grading, not a filtration, on our algebras of operators. When operators act on objects on vector bundles, we obtain a bi-grading.
